Rijksmuseum director Wim Pijbes addresses the world's press in front of Rembrandt's The Night Watch in the Gallery of Honour. Photograph: Peter Dejong/AP

The grand refurbishment of the Dutch national museum of art and history, the Rijksmuseum – home to thousands of masterpieces including that greatest of Rembrandts, The Night Watch – started life as a millennium project.

In the event, the museum closed its doors in 2003 and was due to reopen by 2006. But, dogged by innumerable delays, including a catastrophic tendering process, problems with asbestos and protests by cycling campaigners, it has been shut for a decade, losing an exhausted director along the way.

Now, finally, the €375m (£318m) overhaul is virtually complete and – amid huge national anticipation and excitement – the museum is due to reopen its doors to the public on 13 April, after an inauguration by Queen Beatrix, one of her last engagements before she abdicates at the end of the month. In a new era of economic uncertainty, and spending cuts to Dutch culture, the project feels like it will be one of the last of its kind for many years.

On a chilly Amsterdam morning, gardeners were still busily laying turf in the museum gardens as an enormous digital clock above the entrance counted down the nine days before the royal visit.

Inside, the magnificent galleries smelled of fresh paint and new carpets, signalling the complete renewal of every aspect of the museum.

Only the position of The Night Watch, the "altarpiece" of the museum, according to Wim Pijbes, the museum's director since 2008, has remained the same. Everything else – the architecture, curatorial philosophy and display – has been rethought. The long-awaited results are so spectacular that the museum looks likely to prove a model to other institutions for years to come.

The architects, Spanish practice Cruz and Ortiz, have attempted to return the ornate building – built in 1885 by Pierre Cuypers – back to some of its original splendour. Over the years, many of its elaborate frescoed walls and terrazzo floors had been painted over or removed, and galleries divided up into smaller, white-walled spaces. Now, heavily decorated floors and walls have been reinstated – "a whole new generation of craftspeople have been trained to do it," said Taco Dibbits, the museum's director of collections.

What defeated the architects and the museum directorship was, in the end, a plan to block a cycle path that runs right through the museum, separating its east and west wings. The Fietsersbond (Dutch Cyclists' Union) objected to the loss of the path – and won. According to Dibbits, "The bicycle is folkloric in the Netherlands. Touch the bicycle, and you touch freedom."

Now, to access each end of the ground floor (site of the 18th- and 19th-century collections) visitors must either go via the below-sea-level basement – fruit of a hugely complex engineering project, where an airy new entrance hall, auditorium, cafe and shop have been created – or via the galleries above, which house matchless Rembrandts, Vermeers, Steens and De Hoochs.

Pijbes is philosophical about the sacrifice, but voiced concerns about the potential clash of art-lovers with fast-moving two-wheeled commuters. "We might have 2 million visitors per year, and I am concerned about safety and security. If safety becomes an issue, the bikes have to leave."

Model of 74-gun Dutch battleship William Rex displayed in 17th century gallery of the Rijksmuseum. Photograph: Michael Kooren/Reuters

For the new displays, the rulebook was thrown out. Traditionally, the collection, which contains fine and applied art from the medieval to the modern era, with an emphasis on masterpieces of the Dutch golden age, was shown according to curatorial department. Sculpture was in the sculpture section, ceramics in the ceramics galleries, and so on.

A new approach, however, sees the whole collection, regardless of object type, laid out in a single chronological sweep, with each section devoted to a different century. In a gallery devoted to the young Rembrandt, for example, examples of his early work, including his ravishing self-portrait of 1628-9, are set near finely worked glass and silver by makers Rembrandt knew, furniture of the period, and a portrait by his friend Jan Lievens of Constantijn Huygens, an important patron of the arts. "You get a sense of the world in which Rembrandt was producing his art," said Dibbits.

Though 8,000 objects are on show, the philosophy for the curators when selecting the display was "less is more" – in keeping, said Dibbits, with the often simple aesthetic of Dutch craftsmanship, pointing out the plain, almost modernist-looking pewter communion vessels that had once adorned Amsterdam's greatest church. Curators had to ruthlessly edit their selections for the display ("kill their darlings," as Dibbits put it) to create an atmosphere that was uncluttered and allowed visitors to "appreciate each work of art", he said. "We wanted a display that was about time and beauty," he added.

The objects are shown against walls in a severe palette of shades of grey; and the latest non-reflective ultra-transparent glass has been used for display cases, allowing visitors to appreciate the full effect of seeing, say, 17th-century bronzes set next to mannerist paintings, or medieval wooden devotional sculptures juxtaposed with early Flemish painting.

Labels are concise – 60 words maximum – and the galleries are free of computers and screens. As Duncan Bull, senior curator of foreign paintings, pointed out the subtle connections between a display of Italian paintings, bronzes and ceramics, meant to conjure up the atmosphere of a Renaissance studiolo, he said: "We want people to educate themselves by hints, not to preach at them."

According to Dibbits, the approach of British Museum director Neil MacGregor's Radio 4 series, A History of the World in 100 Objects, buoyed up his confidence in the ability of the artefacts to tell their own stories. "I believe in the authencity, the authority of the objects themselves," he said. "If people want more information, there is plenty they can download on to their smartphones," he added.

Equally, much of the collection has been digitised and is available on the museum's website, "for anyone to use, reuse – and to fire inspiration for artists, makers, scholars and students," said Pijbes. "We want to open the institution up – so that it's no longer authoritarian, a ministry of good taste."

Delftware vases are on display in the 17th century gallery of the Rijksmuseum in Amsterdam. Photograph: Michael Kooren/Reuters

While he wants every Dutch schoolchild to come in and be able to see that national icon, The Night Watch, the entry fee for those over 18 is a hefty €15 (£13) – necessary, said Pijbes, to balance the books; the museum expects to make €10-12m (£8-10m) per year from ticketing.

Steep cuts to Dutch culture funding mean that the museum has had to seek alternative revenue streams. A decade ago 70% of its revenue funding came from the government; now just 50%. "At least," said Pijbes, "it gives me more freedom." Meantime, between the Rijksmuseum and the newly refurbished Stedelijk Museum of modern and contemporary art, "the government has invested over €500m in the city's cultural institutions," he said.

None the less, he added, "My dream is to have free access, as there is to UK national museums, for our national collection to be able to share it to the rest of the world." For now, that looks like an unrealisable ambition.

Five to see at the Rijksmuseum

• Rembrandt's The Night Watch. The most celebrated artwork in Dutch culture is the culmination of the magnificent Gallery of Honour, the lofty hall that brings together the greatest works of the Dutch golden age, including its Vermeers.

• Whalers' woolly hats. Found in 1980 when archaeologists excavated the graves of 17th-century Dutch whalers who had died on Spitsbergen. The hats are shown next to a 17th-century painting of a whale-oil refinery on the Norwegian island.

•Triptych with the Adoration of the Golden Calf. Set in a gorgeous ebony frame, Lucas van Leyden's 1530 masterpiece relegates the biblical scene to the background. In the foreground is a riotous scene of smoking, drinking and partying.

• The Rijksmuseum library. Open for the first time to the public, the library is a balconied four-level space stuffed floor to ceiling with art books, accessible via a perilous spiral staircase. Not for those with a fear of heights.